addresability vs address space vs address bus

Question

How do you determine addressability based on address space? How do you determine the size of the address bus based on the addressability? Ex. The addressability of a machine is 32 bits, what is the size of the address bus?

Answer 1

The address bus connects the CPU with the main memory. So if the address bus has 32 bits, the max size of the main memory is 2^32 bytes, i.e. 4 GB.
The address bus transfers a physical address, and thus the physical address space in this example is 4 GB. However the CPU generates virtual addresses, and the virtual addresses are the virtual address space. The virtual addresses have to be mapped to physical addresses by a memory management unit. In principle, one can map a small virtual address space to a large physical one (as done earlier e.g. in the PDP11 computers), but nowadays mostly a larger virtual address space is mapped to a smaller physical one, e.g. from a 64-bit CPU with a 2^64 byte virtual address space to a physical memory with a 32-bit address bus, which is thus 4 GB large.
So if you have a primitive system without memory management, and you want that all addresses that the GPU can generate are existing main memory addresses, then you address bus must have the same number of bits as the CPU uses for addressing, e.g. 32 bits.
But in a real system the virtual CPU addresses are essentially independent from the physical memory addresses.